## 2. Formulae for dust-driven windsThe optical depth is proportional
to the column density and dust-to-gas mass ratio
. The column density is proportional
to the total mass density and inner
radius of the dusty CSE. The dust
grains in the dust-formation zone are assumed to be in radiative
equilibrium with the incident stellar radiation field, keeping the
effective temperature and the dust
condensation temperature fixed. Then
, the stellar luminosity. The
continuity equation yields , with
gas+dust mass-loss rate , and
expansion velocity . Here the drift
velocity - the velocity difference between the gas and dust fluids -
is neglected, i.e. gas and dust are assumed to be well coupled (cf.
Lamers & Cassinelli 1999). Gas pressure and wind-driving
mechanisms other than radiation pressure on dust are neglected as
well, although they become important for wind speeds below a few km
s The constant of proportionality includes a factor , the wavelength-dependent opacity of the dust, and a temperature dependence as . In a radiation-driven outflow the matter-momentum flux is coupled with the stellar photon-momentum flux via the momentum equation (Gail & Sedlmayr 1986) with the flux-weighted optical
depth. In general, the ratio of and
depends on the mass-loss rate.
Detailed computations such as those presented in Habing et al. (1994),
however, show that this dependence vanishes for
yr Eq. (2) can also be used to eliminate from Eq. (1), yielding a relation between the mass-loss rate and dust-to-gas ratio, and the optical depth and luminosity: Alternatively, Eq. (2) can be used to eliminate For magellanic stars luminosities are easier to measure than expansion velocities, and for these stars it is advantageous to make use of Eq. (4). For galactic stars the opposite is true, and for them Eq. (5) is the formula to use. The constants in Eqs. (4) and (5) are related. Calling the constants of proportionality of Eqs. (1) and (2) respectively and , the constant in Eq. (4) equals and the constant in Eq. (5) equals . The constant of proportionality in Eq. (3) equals . The values of and depend on the properties of the dust species, and calibrating them is an important yet extremely difficult task that will not be exercised here. © European Southern Observatory (ESO) 2000 Online publication: January 31, 2000 |